In a last-ditch attempt to overturn the award of the west coast rail franchise to FirstGroup, Virgin Trains co-owner Sir Richard Branson has offered to run the service "for free" to allow time for parliamentary scrutiny of the decision.

Virgin also renewed its threat to seek a judicial review before the deadline of Tuesday night when contracts are due to be signed by the Department for Transport (DfT), although it hopes pressure from within the Conservative party may sway the transport secretary, Justine Greening, where Labour voices and public petitioning have failed.

The DfT said it had "no reason" to delay signing off the franchise, which will give FirstGroup, Britain's largest rail operator, the right to run services between London and cities north to Glasgow until at least 2026. FirstGroup bid £5.5bn, according to the DfT, £700m more than Virgin, the incumbent and closest shortlisted bidder.

The actual payments will top £10bn after inflation, more than £1bn above Virgin's offer. Branson and several City analysts have questioned whether the level of FirstGroup's bid is sustainable or if risks a repeat of high-bidding franchisees not seeing out their contracts.

Branson on Sunday called for an independent audit of the decision, particularly to check the forecasts of customer numbers, revenue and payments offered to the government. Writing in the Sunday Telegraph, he said: "If this process means extending the current franchise beyond December for a few months, I and my partners at Stagecoach would happily run the extended franchise on a not-for-profit basis, or donate profits to charity. We must ensure that this crucial decision is taken with all the facts correctly assessed and understood."

A DfT spokesman said: "We note the offer that one of the bidders appears to have made via the press. However, the winning bidder was decided by a fair and established process and no reason has been advanced to convince DfT not to sign the agreement."

A Virgin source said it had been making its case at "all levels" of the Conservative party, and MPs were making representations to Greening. The source said: "When we lost CrossCountry to Arriva in 2007 we lost fair and square, by a narrow margin. This is a different ball game altogether, a margin verging on 15% with all the money coming in at the end."

The offer to run it for free meant that there was "no rush, and no need for it to be signed off in August. Let it be properly scrutinised by parliament and we'll abide by the decision."

An online petition started by a Scottish rail passenger calling on the DfT to reconsider has garnered more than 128,000 signatures, a number that all but guarantees a parliamentary debate, although it is unlikely to be held until long after the contract is signed. The transport select committee is planning to question Greening soon over the agreement after its own chair's attempts to delay the award to allow scrutiny were rebuffed.

The Virgin source said: "If this had just been Richard ranting, it would have died off by now. But these are not acolytes – these are people looking independently saying, we don't believe these figures."

FirstGroup is confident the franchise will be formalised by Wednesday and views Branson's reaction as confirmation that the railway line is lucrative enough to support the high premium payments it has promised to pay the government. A source close to the company said: "Our deal represented the best deal for the British taxpayer. We don't have a history of mobilising celebrities, but we do have history of running successful railways."

He said Branson's offer to keep running the line for a short period was "neither practical nor feasible" and added: "We fully expect the process to be completed this week with the signing of the franchise and we look forward to running west coast trains on December 9."